I'm not sure what is meant by "partitioning the <em>x</em>-axis" - partition is a term more often used with computing Riemann sums. So either partitioning has a special meaning for you, or you have to approximate the area with a Riemann sum.
I'm going to assume you just want the exact area of the given region. Notice that the two curves <em>y</em> = 4<em>e</em> ˣ and <em>y</em> = -<em>x</em> + 4 intersect when <em>x</em> = 0 at the point (0, 4). Both curves then meet the vertical line <em>x</em> = 4. The exponential function is increasing while the linear one is decreasing, so 4<em>e</em> ˣ ≥ -<em>x</em> + 4.
The region is then the set of points,
<em>R</em> = {(<em>x</em>, <em>y</em>) | 0 ≤ <em>x</em> ≤ 4 and -<em>x</em> + 4 ≤ <em>y</em> ≤ 4<em>e</em> ˣ}
so the area is given by the integral,
Answer:
The standard form of the equation of the ellipse is
Step-by-step explanation:
The standard form of the equation of an ellipse with center (h , k) is
, where
- The coordinates of the vertices are (h , k ± a)
- The coordinates of the co-vertices are (h ± b , k)
- The coordinates of the foci are (h , k ± c), where c² = a² - b²
∵ The vertices of the ellipse are (-3 , 7), (-3 , 1)
∴ h = -3
∴ k + a = 7 ⇒ (1)
∴ k - a = 1 ⇒ (2)
- Add (1) and (2) to find k
∴ 2k = 8
- Divide both sides by 2
∴ k = 4
- Substitute the value of k in (1) or (2) to find a
∵ 4 + a = 7
- Subtract 4 from both sides
∴ a = 3
∵ The co-vertices of the ellipse are (-5 , 4), (-1 , 4)
∴ k = 4
∴ h - b = -5 ⇒ (1)
∴ h + b = -1 ⇒ (2)
∵ h = -3
- Substitute the value of h in (1) or (2) to find b
∴ -3 + b = -1
- Add 3 to both sides
∴ b = 2
∵ The standard form of the equation of the ellipse is
- Substitute the values of h, k, a, and b in the equation
∴
∴
The standard form of the equation of the ellipse is
Answer:
yeah that's right
Step-by-step explanation:
Answer:
In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. ... Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line.
Answer:
In ∆ABC AND ∆DEF
ABC=DEF...........each 90°
SIDE AB =SIDE ED...........given
SIDE BC =SIDE EF............B-F-C and E-C-F
∆ABC =∆DEF....................by SAS test